Slant Helices Research Articles

Directional spherical indicatrices of timelike space curve

In this work, the directional spherical indicatrices of a timelike space curve using tangent, quasi-normal and quasi-binormal vectors with q-frame are introduced. Then we work on the condition, that a timelike space curve to be slant helix, by using the geodesic curvature of the directional normal spherical indicatrix. Finally, an application of the results is given.

Non-lightlike constant angle ruled surfaces in Minkowski 3-space

In this paper, non-lightlike ruled surfaces of constant slope in Minkowski 3-space with a non-lightlike base curve c(s)=∫αt+βn+γbds, where t, n, b are tangent, principal normal and binormal vectors of an arbitrary timelike curve Γ(s) are investigated. Non-lightlike ruled surfaces of constant slope parallel to a tangent of a timelike general helix and normal of timelike slant helix are studied. It is shown that all non-lightlike ruled surfaces of constant slope are developable but not minimal surfaces.

Slant helix of order n and sequence of Darboux developables of principal‐directional curves

In this paper, we consider the sequence of the principal‐directional curves of a curve γ and define the slant helix of order n (n‐SLH) of the curve in Euclidean 3‐space. The notion is an extension of the notion of slant helix. We present an important formula that determines if the nth principal‐directional curve of γ can be the slant helix of order n (n ≥ 1). As an application of singularity theory, we study the singularities classifications of the Darboux developable of nth principal‐directional curve of γ. It is demonstrated that the formula plays a key role in characterizing the singularities of the Darboux developables of the nth principal‐directional curve of a curve γ.

Special helices on equiform differential geometry of spacelike curves in Minkowski space-time

In our study, we establish k-type helices and (k,m)-type slant helices for equiform differential geometry of spacelike curves in 4-dimensional Minkowski space E₁⁴ and give some new characterizations for these helices. In this paper, we obtain characterizations equiform differential geometry of spacelike curves in 4-dimensional Minkowski space E$_{1}^{4}$. We establish $k$-type helices for this curves. Also we obtain $(k,m)$-type slant helices for equiform differential geometry of spacelike curves in Minkowski space-time.

Spherical Indicatrices of Involute of a Space Curve in Euclidean 3-Space

In this work, we studied the properties of the spherical indicatrices of involute curve of a space curve and presented some characteristic properties in the cases that involute curve and evolute curve are slant helices and helices, spherical indicatrices are slant helices and helices and we introduced new representations of spherical indicatrices.

Spherical Images of W-Direction Curves in Euclidean 3-Space

In this paper, we study the spherical indicatrices of W-direction curves in three dimensional Euclidean space which were defined by using the unit Darboux vector field W of a Frenet curve. We obtain the Frenet apparatus of these spherical indicatrices and the characterizations of being general helix and slant helix. Moreover we give some properties between the spherical indicatrices and their associated curves.

Characterizations of Some Special Curves in Lorentz-Minkowski Space

In a theory of space curves, especially, a helix is the most elementary and interesting topic. A helix, moreover, pays attention to natural scientists as well as mathematicians because of its various applications, for example, DNA, carbon nanotube, screws, springs and so on. Also there are many applications of helix curve or helical structures in Science such as fractal geometry, in the fields of computer aided design and computer graphics. Helices can be used for the tool path description, the simulation of kinematic motion or the design of highways, etc. The problem of the determination of parametric representation of the position vector of an arbitrary space curve according to the intrinsic equations is still open in the Euclidean space E³ and in the Minkowski space <img src=image/13414896_01.gif>. In this paper, we introduce some characterizations of a non-null slant helix which has a spacelike or timelike axis in <img src=image/13414896_01.gif>. We use vector differential equations established by means of Frenet equations in Minkowski space <img src=image/13414896_01.gif>. Also, we investigate some differential geometric properties of these curves according to these vector differential equations. Besides, we illustrate some examples to confirm our findings.

The Bertrand curve associated to a Salkowski curve

After a review of two methods of construction of Bertrand curves, we characterize Bertrand curves defined from Salkowski curves as the only family of Bertrand curves which are slant helices.

A Research on the Spherical Indicatrices of Directional Space Curve

In this paper, we give a parametrization of a directional space curve by using adapted frame called q-frame.. The spherical images of both directional normal indicatrix and directional tangent indicatrix of this space curve with q-frame are studied. Later using the geodesic curvature of the spherical image of the directional normal indicatrices, we work on the condition that a space curve to be slant helix. Finally, we give an application of the results.

Some Associated Curves of Normal Indicatrix of a Regular Curve

In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.

Some Associated Curves of Normal Indicatrix of a Regular Curve

In this paper, we consider integral curves of a vector field generated by Frenet vectors of normal indicatrix of a given curve in Euclidean 3-space. We define some new associated curves such as evolute direction curves, Bertrand direction curves and Mannheim directon curves of the normal indicatrix of a regular curve, respectively. We also found the relationships between curvatures of these curves. By using these associated curves, we give a new approach to construct slant helices and C- slant helices. Finally, we present some examples.

Integrability for the Rotation Minimizing Formulas in Euclidean 3-Space and Its Applications

We analyze integrability for the derivative formulas of the rotation minimizing frame in the Euclidean 3-space from a viewpoint of rotations around axes of the natural coordinate system. We give a theorem that presents only one component of the indirect solution of the rotation minimizing formulas. Using this theorem, we find a lemma which states the necessary condition for the indirect solution to be a steady solution. As an application of the lemma, the natural representation of the position vector field of a smooth curve whose the rotation minimizing vector field (or the Darboux vector field) makes a constant angle with a fixed straight line in space is obtained. Also, we realize that general helices using the position vector field consist of slant helices and Darboux helices in the sense of Bishop.

Darboux helices in three dimensional Lie groups

In this paper, we introduce Darboux helices in a three dimensional Lie group G with a bi-invariant metric and give some characterizations of Darboux helices. Besides, we give some relations between some special curves (general helices and slant helices) and Darboux helices. Moreover, we prove that all Darboux helices are not a slant helix if G is commutative.

K-type hyperbolic slant helices in H3

In the present paper, we give the notion of k-type hyperbolic slant helices in H3, where k 2 {0,1,2,3}. We give the necessary and sufficient conditions for hyperbolic curves to be k-type slant helices in terms of their hyperbolic curvature functions.

Natural Mates of Non-Null Frenet Curves in Minkowski 3-Space

In this paper, we give the definition of the natural mate of a non-null Frenet curve in Minkowski 3-spaces. The main purpose of this paper is to prove some relationships between a non-null Frenet curve and its natural mate. In particular, we obtain some necessary and suffcient conditions for the natural mate of a non-null Frenet curve to be a slant helix, a spherical curve, or a curve of constant curvature. Several applications of our main results are also presented.

Slant helices according to type-2 Bishop frame in Weyl space

Slant helices according to type-2 Bishop frame in Weyl space

Slant helices according to type-1 Bishop frame in Weyl space

Slant helices according to type-1 Bishop frame in Weyl space

Special Curves According to Bishop Frame in Minkowski 3-Space

Abstract Pseudo null curves were studied by some geometers in both Euclidean and Minkowski spaces, but some special characters of the curve are not considered. In this paper, we study weak AW (k) – type and AW (k) – type pseudo null curve in Minkowski 3-space E 1 3 [E_1^3 . We define helix and slant helix according to Bishop frame in E 1 3 [E_1^3 . Furthermore, the necessary and sufficient conditions for the slant helix and helix in Minkowski 3-space are obtained.

(k,m)-type slant helices for partially null and pseudo null curves in Minkowski space 𝔼14{\rm{\mathbb E}}_1^4

Abstract In this study we define the notion of (k,m)-type slant helices in Minkowski 4-space and express some characterizations for partially and pseudo null curves in 𝔼 1 4 {\rm{\mathbb E}}_1^4 .

Persistent rigid-body motions on slant helices

This paper reviews the persistent rigid-body motions and examines the geometric conditions of the persistence of some special frame motions on a slant helix. Unlike the Frenet–Serret motion on general helices, the Frenet–Serret motion on slant helices can be persistent. Moreover, even the adapted frame motion on slant helices can be persistent. This paper begins by explaining one-dimensional rigid-body motions and persistent motions. Then, it continues to present persistent frame motions in terms of their instantaneous twists and axode surfaces. Accordingly, the persistence of any frame motions attached to a curve can be characterized by the pitch of an instantaneous twist. This work investigates different frame motions that are persistent, namely frame motions whose instantaneous twist has a constant pitch. In particular, by expressing the connection between the pitch of Frenet–Serret motion and the pitch of adapted frame motion, it demonstrates that both the Frenet–Serret motion and the adapted frame motion are persistent on slant helices.